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The authors considered robust optimization for polynomial optimization problems where the uncertainty set is a set of possible distributions of the parameter. In specific, this set is a ball around a density function estimated from data samples. The authors showed that this distributionally robust optimization formulation can be reduced to a polynomial optimization problem, hence computationally the robust counterpart is of the same hardness as the nominal (non-robust) problem, and can be solved using a tower of SDP known in literature. The authors also provide finite-sample guarantees for estimating the uncertaity set from data. Finally, they applied their methods to a water network problem.